Combinatorial Algebraic Geometry pp 113-132 | Cite as

# Equations of \(\overline{M}_{0,n}\)

## Abstract

We study the moduli space \(\overline{M}_{0,n}\) of genus 0 curves with *n* marked points. Following Keel and Tevelev, we give explicit polynomials in the Cox ring of \(\mathbb{P}^{1} \times \mathbb{P}^{2} \times \cdots \times \mathbb{P}^{n-3}\) that, conjecturally, determine \(\overline{M}_{0,n}\) as a subscheme. Using *Macaulay2*, we prove that these equations generate the ideal for 5 ≤ *n* ≤ 8. For *n* ≤ 6, we also give a cohomological proof that these polynomials realize \(\overline{M}_{0,n}\) as a subvariety of \(\mathbb{P}^{(n-2)!-1}\) embedded by the complete log canonical linear system.

## MSC 2010 codes:

14H10 (primary) 13D02 (secondary)## Notes

### Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We thank Bernd Sturmfels for providing inspiration and feedback on multiple drafts. We are also grateful to Christine Berkesch Zamaere, Renzo Cavalieri, Diane Maclagan, Steffen Marcus, Vic Reiner, and Jenia Tevelev for many helpful discussions. The second author was partially supported by a scholarship from the Clay Math Institute.

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